Question: Perform the row operation, $R_3+4R_2\rightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} 0 & 0 & 1 & 4 \\ 1 & 2 & 3 & 6 \\ -3 & -3 & -3 & 1 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_2$ and $R_3$ are given below. $R_2=\left[\begin{array} {ccc} 1 & 2 & 3 & 6 \end{array} \right] ~~~~~ R_3=\left[\begin{array} {ccc} -3 & -3 & -3 & 1 \end{array} \right]$ We are asked to perform the row operation, $R_3+4R_2\rightarrow R_3$. Therefore, we must add $4R_2$ to $R_3$. $\begin{aligned}R_3+4R_2 &= \left[\begin{array} {ccc} -3 & -3 & -3 & 1 \end{array} \right] + 4\left[\begin{array} {ccc} 1 & 2 & 3 & 6 \end{array} \right] \\\\&=\left[\begin{array} {ccc}1 & 5 & 9 & 25 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_3$ with $R_3+4R_2$. $\left[\begin{array} {ccc} 0 & 0 & 1 & 4 \\ 1 & 2 & 3 & 6 \\ {-3} & {-3} & {-3} & {1} \end{array} \right] \xrightarrow{R_3+4R_2\rightarrow R_3} \left[\begin{array} {ccc} 0 & 0 & 1 & 4 \\ 1 & 2 & 3 & 6 \\ {1} & {5} & {9} & {25} \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} 0 & 0 & 1 & 4 \\ 1 & 2 & 3 & 6 \\ 1 & 5 & 9 & 25 \end{array} \right]$